ADVANCED ENGINEERING MATHEMATI
ADVANCED ENGINEERING MATHEMATI MATH 348
Colorado School of Mines
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E Kreyszig Advanced Engineering Mathematics 9 ed Section 7179 pgs xxxxxx Lecture Chapter 7 Wrap Up Module 07 Suggested Problem Set Suggested Problems na February 5 2009 Quote of Lecture 7 Chairman Kaga If memory serves me right Iron Chef 19931999 In conclusion of chapter 7 we summarize the important results concerning the linear system of equations Ax b where A E Rm x 6 RM b E Rmxl Speci cally we begin with the following system of linear equations a11x1a12x2a13x3 amxn b1 a211a22I2a233 quot39a2nn 72 1 a311a32I2a333 quotquottaSnIn 73 am1x1am2xzam3xa amnxn 17m which we know has the following matrixVector representation an 112 113 am I1 1 121 122 123 H Mm 2 72 2 Ax 131 132 133 dam is 73 b am am an anm in bm n where the product of matrices is de ned as ABlij Zaikbkj 1 We note that this system can also be k1 de ned as a linear combination of Vectors n 3 AxI1a1I282383quot39n8n ZIjaj b7 where ai E Rm is the 139 column from the coef cient matrix A Given A and b we ask 0 Does there exist a solution to the linear system Ax b with the understanding that there may exist a solution and this solution may be unique Thus we have three possible outcomes when trying to solVe Ax b 1 There exists a solution to Ax b 2 There exists a solution to Ax b and this solution is unique 3 There does not exist a solution to Ax b 1The following website contains an animation of matrix multiplication httpwwwsciwsu edumathfaculty genz220v1essonskent1erFullMultfullMatrixMultiply html We should note that Visually this is not the same way we multiply but it is equivalent This goofy animation httpwwwpurp1emathcommodulesmtrxmulthtm is similar to how we conduct multiplication MATH348 Advanced Eu insulin M quot 2 which can be seen in the 1D case by quick but careful inspection of ax 1 ab E R For the case of coef cient data from Rmxquot we form an augmented matrix 111 112 113 aim b1 121 122 123 Wm 72 4 131 132 ass m dam b3 am am am am bm and apply the rowreduction algorithm2 to as This algorithm makes us of the following three rules for manipulating augmented matrices 1 row scaling the multiplication a row by a nonzero scalar 2 row exchange the exchanges of two rows 3 row replacement the addition of a multiple of one row to another row which we know to be equivalent to algebra applied directly to the linear system of equations We apply this algorithm when taken to full completion in a twopart process In the socalled forward phase we H Begin with the leftmost nonzero column This is a pivot column The inot position is at the top to Select a nonzero entry in the inot column as a pivot If necessary interchange rows to moVe this entry into the inot position Use row replacement operations to create zeros in all positions below the inot E Cover or ignore the row containing the inot position and coVer all rows if any aboVe it apply steps 13 to the submatrix that remains Repeat the process until there are no more nonzero rows to modify While in the backward phase we 1 Begin with the rightmost pivot and working upward and to the left create zeros aboVe each inot If a pivot is not 1 make it 1 by a scaling operation The forward phase produces a row echelon form of the input matrix From this echelon form the backward phase produces the reduced row echelon form of the input matrix3 Since rowoperations do not change the solution to linear systems getting to these echelon forms is the goal of the algorithm For example if we take A E RSXQ and b E 1R6lt1 and reduce it to 1 0 as as 0 as 0 a 0 0 1 0 as as 0 as 0 b 5 0 0 0 1 as as 0 as 0 c O O O O O O 1 as 0 d O O O O O O O O 1 e 0 0 0 0 0 0 0 0 0 f where as are in general nonzero elements then the following simplest linear system whose solution is equiV alent to the solution of Ax b 4 1I1I2I5I6I3 a I3I5I6I3 I 4 Is Is 123 C m m3 d x9 e 0x10x20x3Ox40x50x60x70x30x9 f 2This algorithm is also often called Gaussian elimination in honor of its European inventor Carl Friedrich Gauss httpenwikipediaorgwikiGaussianielimination 3ht p wwwmathaaudk ottosenMatZCrralghtml 4The following website contains an animation of rowireduction httpwwwsciwsuedumathfacultygenz 220VlessonskentlerSolveAnimEchsolveAnimlhtml V There are more at httpwwwsciwsuedumath facui MATH348 Advanced Eu lllccilll M t t39 3 We notice that if f 7 0 then the nal equation is inconsistent and the system has no solution If f 0 then there is a solution to the system but since we started with more columns than rows this solution is not unique 5 From this we notice that not only is it important to deduce from rowreduction the consistency of each equation one must also compare the total number of variables to the number of pivots or the number of free variables This difference determines the uniqueness of the solutions and is an expression of the ranknullity theorem Since the calculation of the differences 1 A1 number of variables number of pivots 2 A2 number of variables number of free variables will be important we record the following de nitions which will allow us to calculate these numbers from rowreduced matrices Before we recite this information we take a minute to note the logic these statements will be used for 1 Given some set of vectors we must determine how to make new ones from vectors from the set linear combination 2 Suppose we make the set of all linear combinations which in general contains an in nite number of vectors we would like a way to specify those vectors necessary for the construction of this spanning set linear independence Spanning sets are examples of socalled linear vector spaces and linearly independent vectors from this setspace constitutes a basis for this space with dimension equal to the number of vectors in any basis E Two vector spaces important in the study of systems involving A are the null space and column space of A The dimension of the null space counts the number of free variables and the dimension of the column space counts the number of pivots Rank Nullity Theorem De nition 1 Linear Combination Let S 3901 v2 v3 vk where vi 6 Rquot for i 1 2 3 k k E N then we say that w E Rquot is a linear combination of the vectors from S if k 6 wc1v1c2v2c3v3ckvk E Cj39lj 71 where 87 E R De nition 2 Linear Independence Let S be as before then we say that S forms a linearly independent set if the following bidirectional implication holds k 7 ZCj39lj0ltgt ci0 foralli123k Remark 1 Recall that 7 is equivalent to Vc 0 where V is a matrix whose jth column is 39le and c is a vector whose elements are 87 If we note without proof that pivot columns are linearly independent then the linearly independent vectors from S are the pivot columns from matrix V and that these columns can be found by the rowereduction applied to V De nition 3 Spanning Set Let S be as before Then we de ne the span of S as the set ofail linear combinations of the the vectors from S That is spanS is the set of all a de ned by De nition 4 Linear Vector Space A linear vector space or just vector space for brevity is a set of vectors S which is also closed under arbitrary linear combinations of the vectors from S6 This is to say that a vector space is a the set S along with all linear combinations of the vectors from S It follows that the span of any set of vectors is a vector space 5In other words the system has more variables than equations and from this underdetermined system one would never expect unique solutions To have unique results one must have at least as many equations as unknowns If there are more equations than unknowns then the system is said to be overdeterrnined 6This de nition is somewhat imprecise There are particular algebraic rules which must hold for the space to be a vector space Please consult 79 of your text for more detailed information MATH348 Advanced Eu lllccllll M t t39 4 De nition 5 Basis Given a vector space say S we say that a basis for this space is the maximum collection of linearly independent vectors from S or equivalently the minimum collection of vectors needed to span the space De nition 6 Dimension Given a vector space S and a basis for this space say BS we say that the dimension of the space is the number of vectors in this basis That is dimB De nition 7 Null Space The null space of a matrix NulA is the vector space de ned byail solutions to the homogenous system Ax 0 Remark 2 If the system Ax 0 is consistent with in nitely many solutions then the general solution will be a linear combination of vectors multiplied by free variables These vectors form a basis for the null space and thus the null space has dimension equal to the number of free variables De nition 8 Column Space The column space of a matrix colA is the set ofail linear combinations of the columns of A Remark 3 From remark 1 we have that the pivot columns of a matrix are linearly independent and thus a basis for the column space of a matrix is its set of pivot columns From this we conclude that the dimension of the column space of a matrix also known as its Rank is the number of pivots in the matrix Theorem 1 RankeNullity Theorem Let A E Rmxquot then the following equality holds 8 RankA dimNulA ri which asserts that the number of pivots plus the number of free variables must be equal to the number of columns in A This summarizes the major concepts from chapter 7 The following statement summarizes this material for the case Where the coef cient matrix is square Theorem 2 The invertible matrix theorem Let A 6 RM Then the following statements are equivae lent 1 A is an invertible matrix That is A71 exists 2 detA 7 O 3 A is row equivalent to the n X 71 identity matrix 4 A has nepivot positions 5 The equation Ax 0 has only the trivial solution 6 The columns of A form a linearly independent set 7 The equation Apb has a unique solution for each I E Rquot 8 The columns of A span Rquot 9 The columns of A for a basis for Rquot 10 colA R 11 dimcolA n 12 rankA n 13 MM a 14 dimnulA O XL39 2x O 5ch L11 qxlquot sz f GX3O MP BXH clth leago Lmkh d 3 Gun 204 ltl lxiiu cu 1 Tc PLv c D I so AHquot mXOIIMuF H JPN4 Ukuno 57 mx w PKuhO W13 674414 uxuno w Er 43 Q95 9X94 H mungjwrn p mgr 4 wur Abra M WX 4 OTJH gonna LerLlqu Zorn 523 MD o OV PM w Mro ivrv 590993 f IA 93 w V4906 EXMEKJ L X LX353 le 535 9x3 1 Ex Ow oxfu 1 3939 0Q M o gwx 15 Own K1242 Tkli 3 cod 703 m W 4 Arak oh ka ms 3amp3 29651 915 H7 Make sea w 3 a M omos wwulxg avg3amp6 I 95wcv A1 P h 39 A sh Wm M rfMsisi39vd EXt negat 1 Kl BXLS X X73 5 3l XL39gtltBO Eltmgt 1 m3gtltn 5 qu QLt 5X3 1 xtir xzfo EXonvaJ 3 X lxlx XSI39i 39XZ X31 x 3gtlt1 O E Kreyszig Advanced Engineering Mathematics 9th ed Section 121 pgs 535538 Lecture Introduction to PDE Module 13 Suggested Problem Set 17 19 22 23 24 26c 27 April 23 2009 Quote of Lecture 13 If you didn t care what happened to me and I didn t care for you We would Zig Zag our way through the boredom and pain occasionally glancing up through the rain Wondering which of the buggers to blame and watching for pigs on the wing Pink Floyd Pigs on the Wing Part 1 1977 1 INTRODUCTION TO PDE At long last we start our study of partial di erential equations PDE We will see everything we have studied this semester come back again as we learn the classical methods of solving linear PDE The emphasis here is on the linearity of the PDE Without this the following important tools 0 Linear combinations of basis vectors 0 Eigenvalues and eigenvectors 0 General solution to linear ODE o Fourier seriesintegral are be rendered almost useless 1 However before we begin that discussion it makes sense to discuss some of the basic terminology De nition 1 Linearity of an Equation 7 We say that an equation di erential or otherwise is linear in some quantity if it can be written as a linear combination of the quantity In the case of a PDE the quantity is the unknown function u which may depend on many variables say x y z t The PDE is then linear if it can be written as a linear combination of u and derivatives of u The general notation can be cumbersome so this is best illustrated by examples Example 1 Linear PDE 7 The following are some examples of common linear PDE 1 Au0 2 3 c2Au 4 82u 2 4 c Au 5 1I say almost because to understand nonlinear theory which is at this time woefully incomplete one must under stand the completeness of linear theory So these principles come back again and again to study nonlinear theory but are incomplete in these sense that donlt often tell you everything you might like to know 2This equation is called Laplacels equation and models space subjected to potential elds like gravitational or electrostatic 3This equation is called Possionls equation and models space subjected to potential elds with source terms 4This equation is called the heat or diffusion equation and models the timeedynarnics of the ow of a density which tends to move from areas of high density to low density 5This equation is called the wave equation and models the standing waves or traveling waves in an elastic medium 6This equation is called a convection equation or transport equation and models the pure transport of a material due to movements of its background medium MATH348 Advanced Eu lllccllll M quot 2 Example 2 Nonlinear PDE e The following are some examples of common nonlinear PDE l pltgvVvgt 7Vp1Av 7 u 2 cAu u 2u 8 Bu Bau Bu 3 a t w t a Remark 1 The critical point here is the for a PDE to be linear you cannot have terms like 09 2 Bu 82u 2 1 u sum um at while terms like 2 u sin are permitted De nition 2 Homogenous PDE e We say that a PDE is homogeneous it is linear and does not contain terms where the dependent variable u or derivatives of this unknown function are absent If the linear PDE contains a term which does not depend on the unknown function or its derivative then we say that the PDE is inhomogeneous Of the previous linear examples I 3 and 5 are homogeneous while is not Theorem 1 Superposition of Solutions 7 If a PDE is linear and homogenous and u1 and u2 are solutions to this PDE then the arbitrary linear combination u mm a2u2 a1 a2 6 R is also a solution Proof In general since the derivative of a sum is the sum of derivatives superpositions will be decomposed by the PDE into smaller equivalent PDE If each term in the superposition is a solution then each smaller equivalent PDE is subsequently satis ed Speci cally for the heat equation we have 3 g a1u1 mm 4 alaitl a2 5 a1c2Au1 a2c2Au2 6 Aa1u1 agug 7 c2Au These arguments hold for Q homogeneous linear equation D This coupled with all of your mathematics up to now completes the basic background necessary for the study of linear PDE which will begin with a derivation of the socalled heat or diffusion equation The heatdiffusion equation in rstorder in time and secondorder in space linear PDE on R3 and models the timedynamics of a conserved density whose ow is along its spatial gradient In this derivation we focus on the generality conservation principles and their closure with constitutive relations The wave equation in R14 can be derived in the context of a vibrating string by analysis of forces at a point on the string However it can manifest more generally in the context of the Einstein eld equations on a vacuum background or in terms of small disturbances of an elastic background medium Though this is outside the scope of our course it is interesting to know since our characterizations of solutions to the wave equation will hold in both contexts and give insight into how nonlinearity might appear Begining with a vibrating elastic rectangular membrane we will derive the solution to the wave equation in RHI by 7This equation is called the Navierestokes equation and can be derived as the model equation for the evolution of uid particles See also httpenwikipediaorgwikiNaviereStokes quations and httpwwwc1aymathorg millenniumNaviereStokesiEquations This equation is called the nonlinear Schrodinger equation and models the evolution of a new phase of matter called a BoseEinstein condensate whose 1995 experimental observation earned researchers at Boulder and MIT and Nobel prize in 2001 See also httpwwwcoloradoeduphysicsZOOObec and httpenwikipediaorgwiki BoseeEinsteinicondensate 9This equation is called the Kortewegide Vries equation an models surface waves in shallow waters and was the basis for modern advances in the study of exactly solvable nonlinear PDE httpenwikipediaorgwiki Kortewege de7Vriesiequat ion MATH348 Advanced Eu lllccllll M quot 3 the use of double Fourier series and moving to a vibrating circular membrane We Will see hoW the solution allows for more complicated vibrational modes Which can be interpreted physically in terms of musical instruments Lastly We characterize solutions to the Wave equation in terms of traveling Waves Whose speed and in uence can be determined by the concept of a characteristics Which gives rise to soundsspeed and the speedof light 2 LECTURE GOAL Our goal With this material Will be 0 Understand the mathematical de nition of PDE as Well as some of their modeling capabilities 3 LECTURE OBJECTIVES The objectives of these lessons Will be 0 List various PDE and their associated models 0 De ne the vocabulary associated With PDE With an emphasis on the interplay between linearity and superposition 0 Outline direction and key points of our study of PDE E Kreyszig Advanced Engineering Mathematics 9 ed Section 119 pgs 518528 Lecture Fourier Transform Module 12 Suggested Problem Set 2 3 9 14a March 31 2009 Quote of Lecture 12 Jenny said when she was just ve years old there was nothin7 happenin7 at all Ehery time she puts on a radio there was nothin7 goin7 down at all not at all Then one ne mornin7 she puts on a New York station you know she don t believe what she heard at all She started shakin7 to that ne ne music you know her life was saved by rock 7n7 roll Despite 7 all the amputations you know you could just go out and dance to the rock 7n roll station and it was alright The Velvet Underground Rock And Roll 1970 We are nally at the end of our study of Fourier methods We have Fourier series to represent periodic functions and Fourier integrals to represent functions which do not necessarily have a periodic feature 1 From the Fourier integral one can then derive the socalled complex Fourier transform or just Fourier transform for short A 1 4W 1 A imac o 7 x 6 dx x 7 o e do f 57wa f 57wa In the last set of notes we mention that these equations have a striking similarity to complex Fourier series and that statements of energy and symmetry have analogies for Fourier transform In the following we use our knowledge of Fourier methods to gather insight into physical process that rely on Fourier analysis Let s rst begin with the following transform pair 1 7w 7 1 L 7 ump aw dti lt2gt3 m76t This statement says if we wish to send a instantaneous pulse of information then its representation in the frequency domain is a constant function We take this to mean the following 1 A completely localized function requires an equal amountamplitude of every possible frequency of oscillation Since the sum of the squares of these amplitudes is proportional to the energy of the timesignal we conclude that this transmission would require an in nite amount of energy This transform highlights a fundamental property of Fourier transforms That is if a function is localized in one domain then it is de localized in the transformed domain This relationship is the basis for the Heisenberg uncertainty principle of quantum mechanics but also has a place any time the Fourier transform concept is used The two most important relations are 0 Positionmomentum In physics position and momentum are related by Fourier transform AxAp Z a E llr and consequently if the position of a quantum particle is highly localized then its momentum is de localized Thus if we know exactly where a particle is then we have no idea about where the particle is going 0 Energytime In physics and engineering energy and time are related by Fourier transform AEAt Z a E llJr and consequently if the event takes place in an in nitesimal amount of time then it requires an in nite amount of energy Though these statements eccentrically highlight important concepts they are motivated with nonrigorous mathematical tricks involving the deltarfunction Maybe a more sensible function is given by ft A 7alttlta 0 otherwise 1Again we mention that periodicity can be reconstructed via delta functionsl Thus making the Fourier integral the more general representation MATH348 Advanced Eu lllccllll M quot 2 This function is commonly called a single nite pulse and can be thought of as a transmission of information A which takes place for 2a units of time The transform of such a function is N which is commonly called a sinc function or sampling function From this transform pair one can gather o f is localized in time f is delocalized in frequency That is some amount of almost every frequency is required to construct the single nite pulse of information Moreover if a a 00 then f is a constant function which we know transforms to a delta function Thus if we consider the limit a gt 00 for a sinc function then we ought to get a delta function 2 o The most dominant contribution to the representation of f comes from the o 0 mode which is to say f is most like7 a constant function but requires the presence of other Fourier modes because it isn t a constant function 3 So why is this called a sampling function That s a good question and is important to shared bandlimited frequency communications down an ideal medium If we assume that f is a signal which possesses a Fourier transform and that this signal is bandlimited in the frequency domain then it is possible using the concept of periodic extension to show that 0 mr sin tL 7 mr Wig 6 which implies that the original signal can be reconstructed using sincsampling functions where the weights of the previous linear combination are given by the original signal sampled every 7rL units in time This result is known as the sampling theorem and from this we conclude o The sinc functions are a basis for all time signals sent out over a frequency limited communication medium That is any signal sent over the radio telephone or cable line can use the previous procedure for mathematical reconstruction 0 To send signals over these communication channels the signal is not needed at every instantaneous moment in time That is it needs to be sampled at integer multiples of 7rL in time where L is de ned to be the cutoff frequency for complete lossless reconstruction 1 LECTURE GOALS Our goals with this material will be 0 Understand the relationship between a function and its Fourier transform as compared to a periodic function and its Fourier coe 39icients o Conceptualize the Fourier transform by applying it to physically motivated systems 2 LECTURE OBJECTIVES The objectives of these lessons will be 0 Calculate the Fourier transform of the delta function and single nite pulse 0 Derive the representation of a signal transmitted via a bandlimited frequency channel 2What a weird thing a delta function7 is In ODE7s you likely considered the limit of rectangles of unit area and now we have the limit of sinc functions There are in fact many more ways to get to a delta functionT 3A constant function wouldn7t truncate for ltl gt a